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Algebra is the language of modern mathematics. This course introduces students to that language through a study of groups, group actions, vector spaces, linear algebra, and the theory of fields.

Week 1 (Lectures 01-03)

Review of linear algebra. Groups. Examples of groups. Basic properties and constructions.

Week 2 (Lecture 04-06)

Permutations. Cosets, Z/nZ.

Week 3 (Lecture 07-09)

Quotient groups, first isomorphism theorem. Abstract fields, abstract vectorspaces. Construction and invariants of vectorspaces.

Week 4 (Lecture 10-13)

Abstract linear operators and how to calculate with them. Properties and construction of operators.

Week 5 (Lecture 14-16)

Orthogonal groups.

Week 6 (Lecture 17-19)

Isometries of plane figures. Cyclic and dihedral groups. Finite and discrete subgroups of symmetry groups.

Week 7 (Lecture 20-22)

Group actions. Basic properties and constructions. Groups acting on themselves by left multiplication. Groups acting on themselves by conjugation.

Week 8 (Lecture 23-25)

A5 and the symmetries of an icosahedron. Sylow theorems. Study of permutation groups.

Week 9 (Lecture 26-28)

Rings. Examples of rings. Basic properties and constructions.

Week 10 (Lecture 29) [Audio Only]

Quotient rings, extensions of rings. Integral domains, fields of fractions.

Week 11 (Lecture 30-32)

Special lecture. (Wow! So descriptive. Umm. I may re-watch these and fill this in later).

Week 12 (Lecture 33-35)

Euclidean domains, PIDs, UFDs. Gauss' lemma. Eisenstein's criterion. Algebraic integers.

Week 13 (Lecture 36)

Structure of ring of integers in a quadratic field. Dedekind domains. Ideal class groups.

Week 14 (Final Two Review Lectures)

Wrap-up.

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Harvard Abstract Algebra
Harvard Abstract Algebra
Harvard Abstract Algebra


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